[FOM] Re: The Myth of Hypercomputation

Aatu Koskensilta aatu.koskensilta at xortec.fi
Tue Feb 17 02:02:23 EST 2004

Piyush P Kurur wrote:

>On Tue, Feb 10, 2004 at 10:11:06PM +0000, Toby Ord wrote:
>	For a "hyper computer" built using a theory say T, one first needs to
>be confident that T is indeed true. For this to be the case, T has to be
>experimentally verified. Suppose that T predicts a fundamental constant
>to have the value $\Omega$ (the halting set coded as a real number), how
>are we going to check it. We consider T to be true iff apart from being
>mathematically consistent should agree to all experiments. All
>predictions made by this theory should be in principle verified. 

Why? Of course, if T is true, then all predictions are in "principle"
verified. The physical theories we have now are not verified in practice 
to such  an extent. The ideal of truth provides guidelines for research, 
but "probable" truth or high  estimated truthlikenss or what you have is 
in practice what is needed for a theory  to be accepted - tentatively - 
as true.

It's entirely  possible that a scenario on the lines of following should
happen. Assume that some physical theory T becomes accepted - say T is 
some form of quantum gravity or string theory or whatnot - and someone 
finds out that some physical set up corresponds to a mechanism
which decides Pi-1 sentences *and* gives counterexamples when the
sentence is not true. We could well have as much confidence in T as in 
any of our now common theories. In addition, we could test the 
predictions about Pi_1 sentences we know to be true, and some we know to 
be false, and see that the results make sense. We could now go
trough interesting undecided Pi-1 sentences until we find one that is 
untrue, and is  proven untrue by the counterexample the physical 
mechanism gives us. If T also  gives us a good idea as to why the 
mechanism works, I don't think it would be sensible  to worry
about philosophical problems about underdetermination of the truth of T
by the finite number of experiments we have tried out, any more than it 
is now for quantum physics, general relativity, etc.

Of course, if this situation should happen, it would only be of interest
if we can actually construct such mechanisms, and if the counterexamples 
were of feasible  size (so that we can for example check them on a 
computer). Uncomputability "round the corner" in some remote part and 
exotic part of the universe could very well be just an "ideal element" 
of the theory in the Hilbertian sense.

Aatu Koskensilta (aatu.koskensilta at xortec.fi)

"Wovon man nicht sprechen kann, daruber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus

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